Answer:
x ≥ 1
Step-by-step explanation:
The function f(x) is defined as increasing on the domain (-8, 4), so the ordering of the arguments is not changed by the function. We can solve the inequality as though f(x) = x, which is increasing everywhere.
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f(4x -3) ≥ f(2 -x²)
4x -3 ≥ 2 -x² . . . . . . using our assumed definition of f(x)
x² +4x -5 ≥ 0 . . . . . subtract 2-x²
(x -1)(x +4) ≥ 0 . . . . factored form; zeros at -4, +1
The values of x that make this true are ones that make the factors have the same signs: x ≤ -4 or x ≥ 1.
We need to make sure that the expressions that are the arguments of f( ) have values that are in the domain of f. For that purpose, we can define the function argument to be "z".
For these values of x, the function arguments are ...
x ≤ -4
z = 4x -3 . . . . argument of f(x)
(z +3)/4 = x . . . . solve for x
(z +3)/4 ≤ -4 . . . . . at the smaller value of x
(z +3) ≤ -16 . . . . . . . . multiply by 4
z ≤ -19 . . . . . . subtract 3
This value of the function argument for f(x) is not in the domain of f(x), so this branch of the solution of the inequality is extraneous.
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x ≥ 1
(z +3)/4 ≥ 1 . . . . . . . from above, an expression for x
z +3 ≥ 4 . . . . . . . multiply by 4
z ≥ 1 . . . . . . . . . subtract 3; this is in the domain of f(x)
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The solution to the inequality is x ≥ 1.
